Abstract

We consider the eigenvalue problem for one-dimensional linear Schrödinger lattices (tight-binding) with an embedded few-sites linear or nonlinear, Hamiltonian or non-conservative defect (an oligomer). Such a problem arises when considering scattering states in the presence of (generally complex) impurities as well as in the stability analysis of nonlinear waves. We describe a general approach based on a matching of solutions of the linear portions of the lattice at the location of the oligomer defect. As specific examples, we discuss both linear and nonlinear, Hamiltonian and -symmetric dimers and trimers. In the linear case, this approach provides us a handle for semi-analytically computing the spectrum [this amounts to the solution of a polynomialequation]. In the nonlinear case, it enables the computation of the linearization spectrum around the stationary solutions. The calculations showcase the oscillatory instabilities that strongly nonlinear states typically manifest.

Received 24 November 2012Accepted 16 April 2013Published online 08 May 2013

Lead Paragraph: We consider the time evolution of a quantum mechanical wave function as governed by the Schrödinger equation. The wave function is distributed spatially on a discrete one-dimensional lattice, i.e., a chain of nodes indexed by integers, so that the spatial derivatives are replaced by differences. The potential function is nonzero only at a few center sites on the lattice, representing either physical impurities or other obstacles such as an external field or a nonlinear material. Since the general solution of the zero potential problem is well-known, we begin by constructing it on the outer (left and right) portions of the lattice. Working our way toward the impurity sites using the restraints of the discrete Schrödinger equation, we find that appropriately defined portions of the outer solution must satisfy a polynomialequation. Our method is also applied to the (again) discrete but (now) nonlinear Schrödinger equation. Here, known stationary solutions are acted upon by a time-dependent perturbation, and we find that appropriately defined portions of the perturbation must satisfy polynomialequations. The main point is to show that the polynomial conditions we derive accurately determine the dynamical stability of the solutions. The success of our method in tracking the associated linear and nonlinear spectra is presented throughout the linear and nonlinear cases. In order to demonstrate the generality of our approach, we show examples using both real valued Hamiltonians and complex parity-time symmetric potentials.